MELJUN CORTES AUTOMATA Formal Language Lecture

8/13/2019 MELJUN CORTES AUTOMATA Formal Language Lecture

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Theory of Automata and

Formal Language

2002

AUTOMATA THEORY PAGE 1

SETSet

It is any well defined collection of objects.

It is a collection of distinct objects, without repetition and without ordering. It is denoted by capital letter.

Example:A = {, ,0 ! = {", #, $% = {a, b, c, d & = {'en, (oseph

Methods in Describing a Set:

". Listing Method) *oster + abular -orm

'isting or enumerating all the elements of a gien set.Example: A = {a, b, c

#. Rue Method) /et0%uilder -orm

Elements hae properties in common. Elements must satisfy a gien rule or condition.

Example: A = {x 1 x is a positie integer 2 3A = {x 1 x 2 4

Ee!ent It is a member of a gien set or an object in the collection.

It is denoted by small letter+s.Example:

A = {a, b, c 5a A5 b A5 c A5 d A

T"#es o$ Set

". %inite) elements can be counted.Example:

A = {a, b, c

#. &n$inite) elements cannot be counted.Example:

A = {x 1 x is a positie integer

Things to re!e!ber in isting ee!ents:

". 6rder of elements is not important#. *epetition must be ignored

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Formal Language

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AUTOMATA THEORY'SetsPAGE (

Subset ) * If A is a subset of %, A is contained in % or eery element of A is in %.

Example:A = {a, b% = {a, b, cA + but + A

,ote:

Eery set is a subset of itself.

An empty set is a subset of eery set.

If A % then there is at least one element in A that is not in %.

Example: 'et A = {", #, $

Subsets o$ A are:P )A*= {{", {#, {$, {", #, {", $, {#, $, {", #, $, {

o chec7:1A1 = #n= 8

Pro#er Subset ) or * A is a proper subset of % if A % and A %

-ardinait" o$ the set ). .* It is the total number of elements in a gien setExample:

A = {", #, $

1A1 = $

E!#t" Set / ,u or 0oid Set It is a set containing no elementExample : A = { or A = {

A = { is not an empty set

Singeton It is a set containing only one element.e.g. A = {"

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Formal Language

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AUTOMATA THEORY'SetsPAGE

-o!#arabe set At least one set is a subset of another set.

!onditions so that A 9 % are comparable:1. A % 9 % A

Example: A = {", #, $, 3% = {3, $, #, "

2. A % 9 % AExample: A = {", #

% = {", #, $

3. A % 9 % AExample: A = {a, b, c

% = {a, b

,on'-o!#arabe Set o set is a subset of the other set. ;A % 9 % A 9 A % " "" > >

ransition -unction;>,x< = >

;>,y< = >

;",x< = "

;",y< = "

T8o T"#es O$ %inite Auto!ata :

". Deter!inistic %inite Auto!ata)D%A*

-or each input symbol in , there is exactly one transition of each state

;possibly bac7 to the state itself, , - = initial state= transition function- = set of final state

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AUTOMATA THEORY'%inite Auto!ata PAGE1


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Formal Language

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Transition Diagra!Examples:". &iagrams that ends with ">.

ransition able

x y> " "" # "# $ "

/trings that can be deried:

"> >">

>"">

>">">

ransition -unction;>,>< = >

;>,"< = "

;",>< = #

;","< = "

;#,>< = >

;#,"< = "

#. &raw a transition diagram that will end with >>".

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AUTOMATA THEORY'%inite Auto!ata PAGE11


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Formal Language

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(> ,on Deter!inistic %inite Auto!ata ),%A*

*RE-ALL: In -A or &-A, for each input there is exactly one transition output of eachstate.

'ets try to modify the finite automaton by alowing ?ero, one or moretransitions from a state on the same input symbol. he modified model iscalled on0&eterministic -inite Automaton.

If for some of L, a, ;,a< does either to a uniue state or seeral statesor not states at all, then the -A is a -A.

A on0&eterministic -inite Automaton is a uintuple K=;L,,>,,- = initial state - = set of final state

It allows ?ero, one or more transitions for eery inputs symbol. Empty string accepted

A seuence of symbols, say a", a#, ...an is accepted by -A if there exists aseuence of transition corresponding to the input seuence that leads fromthe initial states to the final state.

Example:!onstruct a -A that accepts string ending with >"".

;>,>"">>not 3aid

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AUTOMATA THEORY',on Deter!inistic %inite Auto!ata),%A*PAGE 1(


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Formal Language

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6he reason for the word OnondeterministicP is that we are in a state wherethere are multiple outgoing edges all haing the same input symbol ) we hae achoice of next state.

6he only difference between a -A 9 &-A is that in &-A the next statefunction ta7es us to a uniuely defined state whereas in -A the next state ta7es us

to a set of states.

Example : ,%A Diagra!

State Tabe

> "

> {>,$ {>,"

" > {#

# {# {#

$ {3 >

3 {3 {3

Transition %unction

'et the input be >">>" ;>, < = 2

;>, >< = ;;>, < = ;>,>,>< = ;;>,>",>< @

;",>,$@{ =F272

;#, >">>< = ;;>,>">,"< @ ;$,"< @ ;$,>,$ @

{3=F272725

;#, >">>"< = ;;>,"< @ ;$,"< @ ;3,"< = {>," @ { @ {3 = F2721725

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Formal Language

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AUTOMATA THEORY PAGE1

EU&0ALE,-E o$ D%A and ,%A

heorem: -or eery -A model, there exist one and only one euialent &-A

'et K be the euialent &-A of a -A K.K=;L, ,,>,-,",#,{>,",{>,",{>,#,{",#,{>,",#% =#%9={#,{>,#,{",#,{>,",#

Transition Tabe o$ ,%A

> "

> {> {>,"

" {",#

{

# {# {Transition %unction ;{,>< = {

;{,"< = {

;>,>< = >

;>,"< = {>,"

;",>< = {",#

;","< = {

;#,>< = #

;#,"< = {

;;>,"< = {>,> @ (1,0)

= > @ {",# = {>,",#

;;>,"," @ {} = {>,"

;;>,#< = {>,>< @ ;#,>< = {> @ {# = {>,#

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Formal Language

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AUTOMATA THEORY' E2ui3aence o$ D%A and ,%APAGE 15

;;>,#,",# ;;",#< = {",>< @ ;#,>< = {",# @ {# = {",# ;;>,",#< = {>,>< @ ;",>< @ ;#,>< = {> @ {",# @ {# = {>,",# ;;>,",#,"

State Tabe O$ D%A

> "> {> {>,"

" {",# {

# {# {

>," {>,",#

{>,"

>,# {>,# {>,"

",# {",# {

>,",#

{>,",#

{>,"

{ { {

Transition Diagra! O$ D%A

2003

> >,

"

>,",

#

>,#


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Formal Language

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AUTOMATA THEORY PAGE1;

,O,'DETERM&,&ST&- %&,&TE AUTOMAT&O, B&TH I'MO0ES

It is uintuple K = {L,,,>,- where the transition functions maps.

:L x ; @ {< #L

;,a @ 0closure {"= {>,",# @ {",#= {>,",#


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AUTOMATA THEORY' E2ui3aent ,%A o$ ,%A PAGE1@

#. !onert the gien -Ato -A.

- = {$,4 0closure ;>< = {>,",3> = {>= {a,b

,%A

a % E

> { { {",3

" {",# " {

# {$ { {

$ {$ {$ {

3 { {3 {

4 { {4 {

,%A

A b> {",# {",4

" {",# {"

# {$ >

$ {$ {$

3 { {4

4 { {4


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Formal Language

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AUTOMATA THEORY' E2ui3aent ,%A o$ ,%A PAGE1C

- = {$,4

Trans$or!ed ,%A


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Formal Language

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AUTOMATA THEORY PAGE1

REGULAR EJPRESS&O,S

&E-III6: is an alphabet

A regular expression oer is recursiely defined as:

". > is a regular expression denoting the set of {. #. is a regular expression denoting the set of {. $. -or any a, a is a regular expression denoting the set {a.

3. If r,s,t are reg. exp. &enoting sets *,/, respectiely, then;r t s< denotes * u /rs denotes */r denotes *

finite set of symbols

'",'#,': set of strings from

Examples:

a. '"'#= { xyl x'"R S'# is the concatenation of '"9 '#e.g = {a,b

'"= {a,b {}={} '#= {ba,a ={}

'"'#= {a {ba {b {a = {aba,aa,bba,ba

b. '>= { 'i= '' i)"

e.g. ' = {>,"

'>

= {

-ind '$'"= ''>= {>,",{

= {>,"'#= ''"={>,"{>,"

= {>>,>",">,""'$= ''#= {>,"{>>,>",">,""

= {>>>,>>",>">,>"",">>,">","">,"""


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Formal Language

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AUTOMATA THEORY'Reguar E#ressionsPAGE (

KLEE,E -LOSUREof ' is defined as

' = @'i i = >

POS&T&0E -LOSUREof ' is defined as

'T = @'i i = >

Example:' = {>,"

' = {e,>,",>>,"",">",>",">,""",>>>,">",H..'T = {>,",>>,"",">",>",">,""",>>>,">",HH.

' = {"' = {e,","",""","""",H..

Examples: 'et = {>,"

". > = {>

#. >" = {>"$. > T " = {>,"3. " = {e,","",""","""",HHH4. > T " = {e,>,",>>,>>>,HH... ">> = {>>,">>,"">>,""">>,H.C. ;>T",",>>,"",>>>,""",">,>",">",>>>"HHH.8. ;>>T">,","","""",>>>>",">>HH.D. ;>">T">"< = {e,>">,">",">"",">""",">""""HHH..">. ;>T"T"T"


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Formal Language

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AUTOMATA THEORY'Reguar E#ressionsPAGE ((

#. >"

$. >T"

3. "

4. ""T>


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Formal Language

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AUTOMATA THEORY PAGE(

-O,TEJT'%REE GRAMMARS

!ontext0free Urammars describe context0free languages.

*egular set is an example of !ontext -ree language.

!ontext0-ree grammar ;!-U< is a uadruple

U = ;J,,B,/< where J ) set of ariables {/,% ) set of terminals ;a,b,c>",.. J=/,%

={>," /=/

B: / %" % >%

%

1 1

/ %" / %"

1 >%" " >>%"

>>>%" >>>1

>>>"b. ;>T""s ""s

>" """s >" """>s

""">>s

""">>"s

""">>"

""">>"

c. ambn 1 mM= n

/ aKb K aKb 1 aK 1

aabb aaabb

/ aKb / aKb

aaKbb aaKbb


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Formal Language

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aabb aaaKbb

aabb aaabb

aaabb

2003

AUTOMATA THEORY'-ontet %ree Gra!!arPAGE (;

d. / | 0 | 1

/ "s" 1 >s>

11 1

/ >s> / "s" / >s>

>> ">" >>s>>

>> >>">>

e. ?an?bn? 1 nM=>

aabb

/ a/b / a/b

ab aa/bb ab aabb

aabb

f. set of integers ;T or 0 1 H 1 D 1 > 1 H 1 D

1 'C@

/ T / 0

T" 0D T"> 0D8

0D8C

g. {an

bn

cm

dm

1 n,m M > / abcd| aMbcNd

K aKb 1

ab 1

aabbcd aabbccdd

/ aKbcd / aKbcd

aaKbbcd aaKbbccdd

aabbcd aabbccdd


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Formal Language

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aabbcd aabbccdd

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AUTOMATA THEORY'-ontet %ree Gra!!arPAGE (=

or

/ MN

K aKb 1ab

cd 1 cdaabbcd

/ K

aKbcd

aabbcd

h. {xmyxm1 m M= >

/ x/x 1y

" "

/ x/x / x/x xyx xx/xx

xxx/xxx

xxxyxxx


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Formal Language

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AUTOMATA THEORY PAGE(@

PUSHDOB, AUTOMATA

!ontext -ree Urammars machine counterpart.(Just as the RE have anequivalent autometon-FA).

on deterministic deice.(Deterministic version is only a suset o! all "F#$s). Vae input tape, finite control and a stac7

Essentially a -A with control of both input and stac7 O-I'6P list.

StacN ) is a string of symbols from some alphabet. he leftmost symbol of thestac7 is considered to be the OtapeP of the stac7.

MO0ES". An input symbol is used.

&epending on the input symbol, the top symbol on the stac7, and stateof the finite control, a number of choices are possible. Each choiceconsist of next state for the finite control and a ;possibly empty< stringof symbols to replace the top stac7 symbol. After selecting a choice,the input head is adanced one symbol.

2. Input symbol is not used ;-moe). /imilar to the first except that the input symbol is not used and the

input head is not adanced after the moe. Allows the B&A to manipulate the stac7 without reading input symbols.

BAYS TO DE%&,E THE LA,GUAGE A--EPTED +Y PDA no final state ) emptyboth input tape 9 stac7

". /et of all inputs for which some seuence of moes causes the B&A to emptya stac7.

#. &esignate some states as final sates define the accepted language as the set

of all inputs for which some choice of moes causes the B&A to enter finalstate.

with final state ) empty inputtape

K = ;L, , , , o, Wo,-= in L, initial stateWo = in , start symbol- = X L, set of final states= mapping L x ;@ {}) ! to Q x *

;, a, W< = {;B", y"


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Formal Language

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;",>,%UU%%*< ; #, >, %* < ; #, , * < ; #, , < hold the input, which is a string ofsymbols chosen from a subset of the tape symbols called input

symbols. he remaining infinity of cells each hold blan7, which is a special tape

symbol that is not input symbol.


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Formal Language

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-inite !ontrol

+asic Turing Machine

In one moe, the K ;%e&en%in' u&on the symol scanne% y th ta&e hea% thestate o! !inite control)

". changes state#. prints symbol on the tape cell scanned, replacing what was written there$. moes its head left or right one cell

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AUTOMATA THEORY'Turing Machine PAGE

Language Acce#ted

*ecursiely enumerable strings can be enumerated or listed by auring Kachine.

Example: K = ;{o,"H3, {>,", {>,",x,y,%, , >,%, 3>""

/tate Vead Bosition

o >>"">" " x>"">" " x>"">" # x>y">" # x>y">" o x>y">" " xxy">" " xxy">"

# xxyy>" # xxyy>" o xxyy>" $ xxyy>" $ xxyy>" Y

herefore, not accepted


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Formal Language

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AKA !6KB@E* @IJE*/ISBroject 8, Lue?on !ity

!6''EUE 6- !6KB@E* /@&IE/

KA@A'I

A@6KAA VE6*S

Brepared by :MELJUN . !"RTES

K/!/ /@&E


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MELJUN CORTES AUTOMATA Formal Language Lecture

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